On particle collisions in the gravitational field of the Kerr black hole

0.12 MB·English

by  A. A. Grib

On particle collisions in the gravitational field of the Kerr black hole
A.A.Griba,b,Yu.V.Pavlova,c
aA.FriedmannLaboratoryforTheoreticalPhysics,30/32Griboedovcan.,St.Petersburg191023,Russia
bTheoreticalPhysicsandAstronomyDepartment,TheHerzenUniversity,48,Moika,St.Petersburg191186,Russia
cInstituteofMechanicalEngineering,RussianAcademyofSciences,61Bolshoy,V.O.,St.Petersburg199178,Russia
Abstract
ScatteringofparticlesinthegravitationalfieldofKerrblackholesisconsidered. Itisshownthatscatteringenergyofparticlesin
the centre of mass system can obtain very large values not only for extremalblack holes but also for nonextremalones existing
1inNature. Thiscanbeusedforexplanationofstillunresolvedproblemoftheoriginofultrahighenergycosmicraysobservedin
1
Augerexperiment.Extractionofenergyafterthecollisionisinvestigated.ItisshownthatduetothePenroseprocesstheenergyof
0
theparticleescapingtheholeatinfinitycanbelarge.Contradictionsintheproblemofgettinghighenergeticparticlesescapingthe
2
blackholeareresolved.
n
aKeywords: Blackholes,Particlecollisions,Cosmicrays
J

0
31. Introduction The problem of the origin of ultrahigh energy cosmic rays

(UHECR) is a major unsolved issue in astroparticle physics.
c] In [1] we put the hypothesis that Active Galactic Nuclei Surely there can be different mechanisms of getting UHECR
q(AGN)canbethesourcesofultrahighenergyparticlesincos-
from AGN. Our aim is full investigation of the role of pro-
-micraysobservedrecentlybytheAUGERgroup(see[2])due
r cesses of collisions and decays of elementary particles close
gto the processes of converting dark matter formed by super-
tothehorizonoftheKerrblackholesintheAGN.Theseparti-
[heavyneutralparticlesintovisibleparticles—quarks,leptons
clescanbesuperheavyparticlesofdarkmatterorusualprotons,
2 (neutrinos), photons. Such processes as it was discussed pre- ironnuclei,etc.Inthelattercasesforusualparticlestogetlarge
vviouslyin[3, 4]couldleadtotheoriginationofvisiblematter energyonemustconsidermultiplescattering.
6from the dark matter particles in the early Universe when the
5energy of particles was of the Grand Unification (GU) order First calculations of the scattering of particles in the ergo-
7(1013 1014GeV)i.e. attheendofinflationera. sphereof therotatingblackhole, takinginto accountthe Pen-
0 If A−GN are rotating black holes then in [1] we discussed roseprocess,withtheresultthatparticleswithhighenergycan
.
1the idea that “This black hole acts as a cosmic supercollider escapetheblackhole,weremadein[8]. Recentlyin[9]itwas
0
in which superheavy particles of dark matter are accelerated shownthatfortherotatingblackhole(ifitisclosetothecritical
0
1closetothehorizontotheGUenergiesandcanbescatteringin one)theenergyofscatteringisunlimited.Theresultof[9]was
:collisions.” ItwasalsoshownthatinPenroseprocess[5]dark criticizedin[10,11]wheretheauthorsclaimedthatiftheblack
v
matter particle can decay on two particles, one with the nega- hole is nota critical rotating black hole so that its dimension-
i
Xtiveenergy,theotherwiththepositiveoneandparticlesofvery lessangularmomentumA,1butA=0.998thentheenergyis
rhigh energyof the GU order can escape the black hole. Then limited.
a
theseparticlesduetointeractionwithphotonsclosetotheblack
Inthispaperweshowthattheenergyofscatteringinthecen-
holewilllooseenergyanalogouslyuptotheGreisen-Zatsepin-
treofmasssystemcanbestillunlimitedinthecasesofmultiple
Kuzminlimitincosmology[6].
scattering.InSection2,wecalculatethisenergy,reproducethe
Processes of converting dark matter particles into visible
resultsof[9]–[11]andshowthatinsomecases(multiplescat-
ones as well as collisions of dark matter particles close to the
tering)theresultsof[10,11]onthelimitationsofthescattering
horizonoftheblackholeintheAGNcanplayimportantrolein
energyfornonextremalblackholesarenotvalid.InSection2.1
thesolutionoftheproblemofcuspsasitwasdiscussedin[7].
anevaluationofthetimenecessaryforthefreelyfallingparti-
Accordingto the hierarchicalmodelthegalaxieswere formed
cletogettheunboundedlargeenergyincollisionwiththeother
dueto initialinhomogeneitiesofdarkmatterdistribution. But
particleinthecentreofmasssystemismade. Itisprovedthat
after originationof rotatingblack holesat the centre of future
inordertohaveinfinitelylargeenergyofcollisiononthehori-
galaxies large part of the dark matter became redistributed so
zon infinitely large time from the beginning of free falling of
thatitcanbemainlypresentinthehaloofgalaxy[7].
theparticleontherotatingblackholeisneeded. Particlecolli-
sionsinsidetherotatingblackholeareconsideredinSection3.
InSection4,weobtaintheresultsfortheextractionoftheen-
E-mailaddresses:[email protected](A.A.Grib),
[email protected](Yu.V.Pavlov) ergy after collision in the field of the Kerr’s metric. It occurs
thatthePenroseprocessplaysimportantroleforgettinglarger whereui = dxi/ds. Takingthesquared(9)andduetouiu = 1
i
energiesofparticlesatinfinity. Ourcalculationsshowthatthe oneobtains
conclusionof[11]abouttheimpossibilityofgettingatinfinity
E =m√2 1+ui u . (10)
theenergylargerthantheinitialoneinparticlecollisionsclose c.m. (1) (2)i
totheblackholeiswrong. q
Thescalarproductdoesnotdependonthechoiceofthe coor-
ThesystemofunitsG =c=1isusedinthepaper.
dinateframeso (10) is valid in anarbitrarycoordinatesystem
andforarbitrarygravitationalfield.
2. Theenergyofcollisioninthefieldofblackholes We denote x = r/M, x = r /M, x = r /M, A = a/M,
H H C C
l = L /M. Apply the formula(10) for calculation of the en-
n n
Let us consider particles falling on the rotating chargeless
ergyinthecentreofmassframeoftwocollidingparticleswith
black hole. The Kerr’s metric of the rotating black hole in
angular momenta L , L , which are nonrelativistic at infinity
1 2
Boyer–Lindquistcoordinateshastheform (ε = ε = 1) and are moving in Kerr’s metric. Using (1),
1 2
2Mr(dt asin2θdϕ)2 (6)–(8)oneobtains[9]
ds2 =dt2 −
− r2+a2cos2θ E2 1
c.m. = 2x2(x 1)+l l (2 x)
(r2+a2cos2θ) dr2 +dθ2 (r2+a2)sin2θdϕ2, (1) 2m2 x∆x" − 1 2 −
− ∆ !− +2A2(x+1) 2A(l1+l2) (11)
−
where
2x2+2(l A)2 l2x 2x2+2(l A)2 l2x ,
∆=r2 2Mr+a2, (2) − 1− − 1 2− − 2
− q(cid:16) (cid:17)(cid:16) (cid:17)#
Misthemassoftheblackhole,J =aMisangularmomentum. where
Inthecasea = 0themetric(1)describesthestaticchargeless
∆ = x2 2x+A2 =(x x )(x x ). (12)
blackholeinSchwarzschildcoordinates.Theeventhorizonfor x − − H − C
theKerr’sblackholecorrespondstothevalue Tofindthelimitr r fortheblackholewithagivenangu-
H
larmomentumAone→musttakein(11)x = x +αwithα 0
r =rH M+ √M2 a2. (3) and do calculations up to the order α2. TakHing into acco→unt
≡ −
A2 = x x , x +x =2,afterresolutionofuncertaintiesinthe
TheCauchyhorizonis H C H C
limitα 0oneobtains
→
r =r M √M2 a2. (4)
C
≡ − − E (r r ) (l l )2
c.m. → H = 1+ 1− 2 , (13)
Thesurfacecalled“thestaticlimit”isdefinedbytheexpression 2m s 2xC(l1 lH)(l2 lH)
− −
r =r M+ √M2 a2cos2θ. (5) where
0
≡ −
2x 2
The region of space-time between the horizon and the static l = H = 1+ √1 A2 . (14)
H
limitisergosphere. A A −
For equatorial(θ = π/2) geodesicsin Kerr’s metric (1) one Inthe limitr r(cid:16) forthe extr(cid:17)emalblackholeoneobtains
H
→
obtains([13], 61) theexpression
§
dt 1 2Ma2 2Ma l 2 l 2
= r2+a2+ ε L , (6) a= M E (r r )= √2m 2− + 1− , (15)
dτ ∆ r − r ⇒ c.m. → H l 2 l 2
" ! # r 1− 2−
given first in [9], showing the unlimited increasing of the en-
dϕ 1 2Ma 2M
dτ = ∆ r ε+ 1− r L , (7) ergy of collision when the specific angular momentumof one
" ! # offallingparticlesgoestothe limitingpossiblevalueequalto
dr 2 2M a2ε2 L2 ∆ 2M necessary for achieving the horizonof the extremalblack
dτ =ε2+ r3 (aε−L)2+ r2− − r2 δ1, (8) hole.
! Formula(8)leadstolimitationsonthepossiblevaluesofthe
where δ = 1 for timelike geodesics (δ = 0 for isotropic angular momentum of falling particles: the massive particle
1 1
geodesics), τ is the proper time of the moving particle, ε = free falling in the black hole with dimensionless angular mo-
const is the specific energy: the particle with rest mass m has mentumAbeingnonrelativisticatinfinity(ε=1)toachievethe
the energyεm in the gravitationalfield (1); Lm = constis the horizonof the black hole must have angular momentumfrom
angularmomentumoftheparticlerelativetotheaxisorthogo- theinterval
naltotheplaneofmovement.
2 1+ √1+A =l l l =2 1+ √1 A . (16)
Let usfind the energy Ec.m. in the centreof masssystem of − L ≤ ≤ R −
two collidingparticleswiththesame restmin arbitrarygravi- Notethat(cid:16)forthement(cid:17)ionedlimitingvalu(cid:16)estherighth(cid:17)andside
tationalfield. Itcanbeobtainedfrom oftheformula(8)iszerofor
(E ,0,0,0)=mui +mui , (9) x =2 1+ √1 A A, x =2 1+ √1+A +A (17)
c.m. (1) (2) R − − L
2 (cid:16) (cid:17) (cid:16) (cid:17)
forl=l andl=l correspondingly. (see(8)in[11]). SoevenforvaluesclosetotheextremalA =
R L
ThedependenceoftheenergyofcollisionE ontheradial 1 of the rotating black hole Emax/m can be not very large as
c.m. c.m.
coordinate for l = l , l = l (see (16)) is given on Fig. 1. it is mentioned in [10, 11]. So for A = 0.998 considered
1 R 2 L max
TheboldfacelinecorrespondstoA=0.998,theordinarycurve as the maximal possible dimensionless angular momentum of
theastrophysicalblackholes(see[14]),from(18)oneobtains
Ec.m.(cid:144)m Emax/m 18.97.
20 c.m. ≈
Doesitmeanthatinrealprocessesofparticlescatteringinthe
17.5 vicinity of the rotating nonextremalblack holes the scattering
15 energyislimitedsothatnoGrandUnificationorevenPlanck-
ean energiescan be obtained? Let us show that the answer is
12.5
no! If one takes into account the possibility of multiple scat-
10
teringsothattheparticlefallingfromtheinfinityontheblack
7.5 hole with some fixed angularmomentumchangesits momen-
5 tum in the result of interaction with particles in the accreting
disc andafterthisisagainscatteringcloseto the horizonthen
2.5
r thescatteringenergycanbeunlimited.
x1 x2 €r€€€€€€€ Thelimitingvalueoftheangularmomentumoftheparticle
1.1 1.2 1.3 1.4 H
closetothehorizonoftheblackholecanbeobtainedfromthe
condition of positive derivative in (6) dt/dτ > 0, i.e. going
Figure1:Thedependenceoftheenergyofparticlecollisiononthecoordinater.
“forward”intime.Soclosetothehorizononehasthecondition
to A = 0.9, the dotted to A = 0.5, the line formed by points l<ε2xH/Awhichforε=1givesthelimitingvaluelH.
describes nonrotating black hole (A = 0). The fractures on From(8)onecanobtainthepermittedintervalinrforparti-
dense line, denoted by dotted lines correspond to values x = cleswithε=1andangularmomentuml=lH δ.Inthesecond
−
orderinδclosetothehorizononeobtains
x (A)M/r (A)(see x in(17))i.e. tozeroesoftheexpressions
R H R
inrootsinformula(11). δ2x2
l=l δ x. x + C . (20)
PuttingthelimitingvaluesofangularmomentalL,lRintothe H − ⇒ H 4x √1 A2
H
formula (13) one obtainsthe maximalvalues of the energyof −
Iftheparticlefallingfromtheinfinitywithl l arrivestothe
collisionforparticlesfallingfrominfinity R
≤
regiondefinedby(20)andhereitinteractswithotherparticles
Emax(r r )=
c.m. → H oftheaccretiondiscoritdecaysonmorelightparticlesothatit
2m 1 A2+ 1+ √1+A+ √1 A 2 getsthelargerangularmomentuml1 =lH −δ,thendueto(13)
= − − . (18) thescatteringenergyinthecentreofmasssystemis
√41 A2 vut (cid:16)1+ √1 A2 (cid:17)
− − m 2(l l )
ThedependenceofEmaxontheangularmomentumoftheblack E H − 2 (21)
holeisshownonFig.c.2m.. c.m. ≈ √δ s1 √1 A2
− −
anditincreaseswithoutlimitforδ 0. ForA = 0.998and
Ecm.max.(cid:144)m l =l , E 3.85m/√δ. → max
2 L c.m.
Notethatfo≈rrapidlyrotatingblackholesA = 1 ǫ thedif-
−
16 ferencebetweenlH andlR isnotlarge
√1 A
12 l l = 2 − √1 A+ √1+A A
H R
− A − −
8 2(√2 1)(cid:16)√ǫ, ǫ 0. (cid:17) (22)
≈ − →
For A = 0.998, l l 0.04sothepossibilityofgetting
4 max H R
− ≈
smalladditionalangularmomentumininteractionclose tothe
a
€€€€€€€€ horizonseemsmuchprobable.
0.2 0.4 0.6 0.8 1 M Thatiswhywecometoaconclusionthatunboundedlylarge
energycanbeobtainedinthecentreofmasssystemofscatter-
ing particles. This means that physical processes not only on
Figure 2: The dependence of the maximal energy of collision for particles
the Grand Unificationscale buteven on the Planckean energy
fallingfrominfinityontheblackholeangularmomentum.
cangointhiscase.
For nonrotatingblack hole the maximal energyof collision One can obtain the same result of getting unbounded large
dueto(18)isEmax/m=2√5(firstfoundin[12]). energyin the centre of mass frame for particles with different
c.m.
ForA=1 ǫ withǫ 0formula(18)gives: massesm ,m andε ,ε [15].
1 2 1 2
− →
Howeveranimportantquestionistolookhowparticleswith
m m 4,06
Ecm.max.(r→rH)∼2 21/4+2−1/4 ǫ1/4 ≈ ǫ·1/4 . (19) largeenergycanbeextractedfromtheblackhole?Someenergy
(cid:16) (cid:17) 3
surelywillbeloosedduetotheredshift, butaswe’llshowin Fortheintervalofpropertimeofthefreefallingtotheblack
thispaperultrarelativisticparticlesstillcanbeobservedoutside holeparticleoneobtainsfrom(8)
theblackhole.Observationofthemcangiveusinsightintothe
GrandUnificationandPlanckeanphysicsnearthehorizon. x0 √x3dx
Beforeansweringonthisquestionletusinvestigatetheprob- ∆τ= M . (27)
2x2 l2x+2(A l)2
lemaboutthetimeneededfortheenergytogrowunboundedly Zxf − −
large. p
Iftheangularmomentumoftheparticlefallinginsidetheblack
hole is such that l < l < l then the propertime is finite for
2.1. Thetimeofmovementbeforethecollisionwithunbounded L R
x x . ForA=1,l=2theintegral(27))isequalto
energy f → H
Let us show that in order to get the unboundedly growing M √x 1 x0
∆τ= 2√x(3+x)+3ln − (28)
energy one must have the time intervalfrom the beginningof 3√2 √x+1
thefallinginsidetheblackholetothemomentofcollisionalso !(cid:12)(cid:12)(cid:12)xf
(cid:12)
growing infinitely. This is connected with the fact of infinity and it diverges logarithmically when xf 1(cid:12)(cid:12). So to get the
→
of coordinate interval of time needed for a freely particle to collision with infinite energyoneneedsthe infinite intervalof
cross the horizon of the black hole (for Schwarzschild metric ascoordinateaspropertimeofthefreefallingparticle.
thisquestionwasconsideredin[16]). Togetlargefiniteenergyonemusthavesomefinitetime[15].
From Fig. 1 one can see that the collision energy can get So to have the collision of two protonswith the energyof the
large values in the centre of mass system only if the collision orderofthe GrandUnificationonemustwait forthe extremal
occursclosetothehorizon.Unboundedlylargeenergyofcolli- black hole of the star mass the time 1024s, which is larger
∼
sionsoutsideofablackholearepossibleonlyforcollisionson thantheageoftheUniverse 1018s. Howeverforthecollision
≈
horizonx x (see(11),(13)). withtheenergy103 largerthanthatoftheLHConemustwait
H
→
Fromequationoftheequatorialgeodesic(6),(8)foraparti- only 108s.
≈
clewithdimensionlessangularmomentumlandspecificenergy From(7),(8)fortheangleoftheparticlefallinginequatorial
ε = 1(i.e. the particleis nonrelativisticatinfinity)fallingon planeoftheblackholeoneobtains
the black hole with dimensionless angular momentum A one
obtains x0 √x(xl+2(A l)) dx
dr (x x )(x x ) 2x2 l2x+2(A l)2 ∆ϕ= − . (29)
= − H − C − − . (23) (x x )(x x ) 2x2 l2x+2(A l)2
dt − √x px3+A2x+2A(A−l) Zxf − H − C − −
p
So the coordinatetime (propertime of the observerat rest far IfA,0,thenintegral(29)isdivergentforx x . Sobefore
f H
fromtheblackhole)oftheparticlefallingfromsomepointr = collision with infinitelylargeenergythe partic→lemustcommit
0
x Mtothepointr = x M >r isequalto infinitelylargenumberofrotationsaroundtheblackhole.
0 f f H
x0 √x x3+A2x+2A(A l) dx
∆t = M − . (24) 3. Collisionofparticlesinsidetherotatingblackhole
(x x )((cid:16)x x ) 2x2 l2x+(cid:17)2(A l)2
Zxf − H − C − −
p As one can see from formula (11) the infinite value of the
Incaseoftheextremalrotatingblackhole(A=1,x = x =
R H collision energyin the centre of mass system can be obtained
1) and the limiting value of the angular momentum l = 2 the
insidethehorizonoftheblackholeontheCauchyhorizon(4).
integral(24)isequalto Indeed, from (12) the zeroes of the denominatorin (11): x =
x , x= x , x=0.
M 2√x(x2+8x 15) √x 1 x0 H C
∆t = − +5ln − (25) Let us find the expression for the collision energy for x
√2 3(x−1) √x+1!(cid:12)(cid:12)xf xC. Note that the Cauchy horizon can be crossed by the fr→ee
(cid:12)
(cid:12) falling fromthe infinity particle underthe same conditionson
anditdivergesas(xf −1)−1forxf →1. (cid:12)(cid:12) the angular momentum (16) as in case of the event horizon.
In allothercases, when A 1 andangularmomentl < l ,
H Denote
≤
integral(24)divergeslogarithmicallyif x x . Thisfollows 2x 2
f → H l = C = 1 √1 A2 . (30)
fromequality C A A − −
(cid:16) (cid:17)
x3+A2x+2A(A−l)=(x−xH)(x2+xHx+2xH)+2A(lH−l).(26) NotethatlL <lC ≤lR ≤lH.
Tofindthelimitr r fortheblackholewitha givenan-
C
Note that for the particle with ε = 1 outside the horizon for gular momentum A o→ne must take in (11) x = x +α and do
C
A < 1 the angular momentum l < lH (see (20)). So for all calculationswithα 0. Thelimitingenergyhastwodifferent
→
possible values of l and A to get the collision with infinitely expressionsdependingonthevaluesofangularmomenta.If
growing energy in the centre of mass system needs infinitely
largetime. (l l )(l l )>0, (31)
1 C 2 C
− −
4
tih.ee.nl1,l2 areeitherbothlargerthanlC,orbothsmallerthanlC, =µ ε2 + 2M (aε L )2+ a2ε21µ−L21µ ∆
s 1µ r3 1µ− 1µ r2 − r2
Ec.m.(2rm→rC) = s1+ 2xH(l1(l−1−lCl)2()l22−lC). (32)  ε2 + 2M (aε L )2+ a2ε22µ−L22µ ∆ . (39)
− s 2µ r3 2µ− 2µ r2 − r2 
This(flormulla)(ilssimlil)a<rt0o,(13)ifeverywhereH ↔C. If (33) Thesignsin(39)areputsothattheinitialparticlesandthepar-
1− C 2− C ticle(2)goinsidetheblackholewhileparticle(1)goesoutside
i.e. l2 (lL, lC), l1 (lC, lR)(ortheopposite),then the black hole. The valuesε1µ,ε2µ are constantson geodesics
∈ ∈ ([13], 61)sotheproblemofevaluationoftheenergyatinfinity
§
E A 2(l l )(l l ) extractedfromtheblackholeincollisionreducestoaproblem
c.m. = 1− C C − 2 , x xC.(34) tofindthesevalues.
m 1− √1−A2 s √1−A2(x−xC) → Theinitialparticlesinourcaseweresupposedtobenonrela-
Itisseen thatthelimitisinfinite forallvaluesofangularmo- tivisticatinfinity:ε1 =ε2 =1,soEq.(39)becomes
menta l ,l (33). This could be interpreted[17] as the known
1 2
instability of the internal Kerr’s solution (see [18, chap. 14]). m 2M (a L )2+ 2M L21
However,fromEq.(6)wecansee − s r3 − 1 r − r2
ddτt(r→rC+0)=( −+∞∞,, iiff ll<>llCC., (35) + s2rM3 (a−L2)2+ 2rM − Lr222 
Tizheadt(isseweahlysoth[e17c]o)l.lisionswithinfiniteenergycannotbereal- =µ ε2 + 2M (aε L )2+ a2ε21µ−L21µ ∆
s 1µ r3 1µ− 1µ r2 − r2
4. STchhewaerxztsrcahcitldio’nsanodfKeenrre’rsgmy etarfitcesr the collision in  2M a2ε2 L2 ∆
ε2 + (aε L )2+ 2µ− 2µ . (40)
− s 2µ r3 2µ− 2µ r2 − r2 
mwieLtnhetatmuLass,sceLosnsmfiad,lelsirnpgtehcieinfitcocasaeenbewlarhgckeienhsoiεnl1ete,orεac2cc,tuisrorpneedcbifefiotwcrseaoenmngueplararr>timcrleos-. Consideratfirstthecasea=0ofthenonrotatingblackhole.
1 2 H InthiscasetheEq.(40)hastheform
Let two new particles with rest masses µ appeared, one of
them(1)movedoutsidetheblackhole,theother(2)movedin- 2M L2 2M 2M L2 2M
sideit. Denotethespecificenergiesofnewparticlesasε ,ε , m 1 1 + 2 1
tt—hoeriitarhleapinrlga4un-levaeorlfmotchoiemtiereosn.ttaaCti(oningnsbuidlnaeicrtskpohafrotµliec).leasmLo1vµe,mLe2µn,tivnit=he1dµexqi/u2daµs- − s r − r=2 µ − εr2 ! 1s+ Lr21µ−1r2 2M− r !
ergCyomanns(deuri(m1v)ao+tmioueni(n2)lt)auw=msµlie(navdi(i1n)teo+lavsi(t2i)c).particlecollisionsfor the(3e6n)- vtε21µ−1+ L22rµ2 1 −2Mr !. (41)
Equamtio(εn1s+(3ε62))fo=rµt(aεn1dµ+ϕ-εc2oµm),ponentscanbewrittenas (37) T4Mhe(mseaex(im16a)l,e(1n8er)g).y−Forvtfocmo2lEµliq−s.io(n41i)sofron2reLg 1et=s−f−orLr2L,!|L1=| =L|L2|(i=n
1µ 2µ
m(L +L )=µ(L +L ), (38) particular,fortheradialmovementoftheprodu|ctso|fre|acti|on)
1 2 1µ 2µ
ε ε . The initial particles were supposed to be nonrela-
1µ 2µ
≤
i.e. thesumofenergiesandangularmomentaofcollidingpar- tivistic oninfinity andso from(37) one getsforthe energyof
ticlesisconservedinthefieldofKerr’sblackhole. theparticlemovingoutsidetheblackhole
Forther-componentfrom(8)oneobtains
µε m. (42)
1µ
≤
2M a2ε2 L2 ∆ Forradialmovementtheenergyradiatedtoinfinityoftheparti-
m ε2+ (aε L )2+ 1− 1
− s 1 r3 1− 1 r2 − r2 clescollidedintheSchwarzschildblackholecannotbelarger
+ sε22+ 2rM3 (aε2−L2)2+ a2ε22r2−L22 − r∆2  t|tLhhe2aN|nno=rttheeae4tcMhrteiao,sttneLeqon1ucµeacrlu=gitryysLoio2nfnµo(t4=nh2ee)0idins(oittouhibabeltlaerpaidnadreSitadicclhilnmwe!coaarvszeesmcLhe1inl=dt),r−aifLd2iru,s=|.L14|M=
 5
For the case when the collision takes place on the horizon x=r/M 1
→
oftheblackhole(r r )thesystem(37)–(40)canbesolved
exactly → H vt(1) = vϕ(1) = α + l1µ , (44)
ut uϕ x 1 2
ε =AL1µ, ε =2m AL1µ, L =m(L +L ) L . (43) (1) (1) −
1µ 2rH 2µ µ − 2rH 2µ µ 1 2 − 1µ vr(1) = 2α2 + 2l1µα + 3l2 1. (45)
In general case the system of three Eqs. (37)–(40) for ur −s(x 1)2 x 1 8 1µ− 2
(1) − −
four variables ε , ε , L , L can be solved numeri-
1µ 2µ 1µ 2µ
cally for a fixed value of one variable (and fixed parameters Duetotheconditiondt/dτ>0(movementforwardintime)the
m/µ, L /M, L /M, a/M, r/M). The example of numerical necessary condition for collinearity is that both (44) and (45)
1 2
solution is µ/m = 0.3,l = 2.2, l = 2.198, A = 0.99, x = mustbezero,whichisnottrue.Thisleadstotheconclusionthat
1 2
1.21, l = 16.35, l = 1.69, ε = 7.215, ε = 0.548. the considerationsof the authorsof [11] forscattering exactly
1µ 2µ 1µ 2µ
− −
Notethattheenergyofthesecondfinalparticleisnegativeand onthehorizoncannotbeusedfortherealsituationofparticle
theenergyofthefirstfinalparticlecontrarytothelimitobtained scatteringclosetothehorizon.
in[11]islargerthantheenergyofinitialparticlesasitmustbe Allthisconfirmsourhypothesisin[1]thatthevicinityofthe
in the case of a Penrose process [8]. If the mass of the parti- rotatingblackholeisthearenaoftheGrandUnificationphysics
cleisnotverylargethisparticleisobservedasultrarelativistic. asitwasintheearlyUniverse.IntheearlyUniversegravitation
Whatisthereasonofthiscontradiction? Letusinvestigatethe createdpairsofsuperheavyX,X˜ particlesdecayingthenasshort
problemcarefully. livingandlongliving XS, XL particlesonordinaryquarksand
Note that if one neglects the states with negative energy in leptons. ButthenduetothebreakingoftheGUsymmetryXL
ergosphereenergyextractedin the consideredprocesscannot becamemetastableandsurvivedupto ourtimeasdarkmatter
belargerthantheinitialenergyofthepairofparticlesatinfinity, particles.
i.e. 2m. The same limit 2m for the extracted energy for any In the vicinity of rotating black holes XL can decay on or-
(including Penrose process) scattering process in the vicinity dinary particles as it is possible for the GU energies as well
of the black hole was obtained in [11]. Let us show why this as they can be converted into XS decaying particles because
conclusionisincorrect. XLXS , 0. These decays can go with baryon and CP-
Iftheangularmomentumofthefallingparticlesisthesame hcon|serviationbreaking,sothatdifferenthypothesesonthepro-
then(see(40))onehasthesituationsimilartotheusualdecayof cesses in the early Universe can be checked by astrophysical
theparticlewithmass2mintwoparticleswithmassµ. Dueto observationsofAGN.
thePenroseprocessinergosphereitispossiblethattheparticle
fallinginsidetheblackholehasthenegativerelativetoinfinity
Acknowledgments
energy and then the extracted particle can have energy larger
than2m.
The authors are indebted to Ted Jacobson and Thomas P.
Themainassumptionmadein[11]isthesuppositionofthe Sotiriou for putting their attention to a numerical mistake in
collinearityofvectorsof4-momentaoftheparticlesfallingin- thefirstversionofthepaper[19].
side and outside of the black hole (see (9)–(11)in [11]). The
authorsof [11] say that these vectorsare “asymptoticallytan-
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6
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7